Theorem rdgon 5973

Description: Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.)

Hypotheses

Ref	Expression
rdgon.1	⊢ (𝜑 → 𝐹 Fn V)
rdgon.2	⊢ (𝜑 → 𝐴 ∈ On)
rdgon.3	⊢ (𝜑 → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)

Assertion

Ref	Expression
rdgon	⊢ ((𝜑 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝜑,𝑥

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rdgon

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5178	. . . . 5 ⊢ (𝑧 = 𝑥 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘𝑥))
2	1	eleq1d 2106	. . . 4 ⊢ (𝑧 = 𝑥 → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
3	2	imbi2d 219	. . 3 ⊢ (𝑧 = 𝑥 → ((𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On) ↔ (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On)))
4		fveq2 5178	. . . . 5 ⊢ (𝑧 = 𝐵 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘𝐵))
5	4	eleq1d 2106	. . . 4 ⊢ (𝑧 = 𝐵 → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (rec(𝐹, 𝐴)‘𝐵) ∈ On))
6	5	imbi2d 219	. . 3 ⊢ (𝑧 = 𝐵 → ((𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On) ↔ (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ∈ On)))
7		r19.21v 2396	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On) ↔ (𝜑 → ∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On))
8		rdgon.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ On)
9		fvres 5198	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑧 → ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) = (rec(𝐹, 𝐴)‘𝑥))
10	9	eleq1d 2106	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑧 → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
11	10	adantl 262	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑧) → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
12		rdgon.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)
13		fveq2 5178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
14	13	eleq1d 2106	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ∈ On ↔ (𝐹‘𝑤) ∈ On))
15	14	cbvralv 2533	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ On ↔ ∀𝑤 ∈ On (𝐹‘𝑤) ∈ On)
16	12, 15	sylib 127	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑤 ∈ On (𝐹‘𝑤) ∈ On)
17		fveq2 5178	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) → (𝐹‘𝑤) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)))
18	17	eleq1d 2106	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) → ((𝐹‘𝑤) ∈ On ↔ (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
19	18	rspcv 2652	. . . . . . . . . . . . . 14 ⊢ (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (∀𝑤 ∈ On (𝐹‘𝑤) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
20	16, 19	syl5com 26	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
21	20	adantr 261	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑧) → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
22	11, 21	sylbird 159	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑧) → ((rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
23	22	ralimdva 2387	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → ∀𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
24		vex 2560	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
25		iunon 5899	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ ∀𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On) → ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On)
26	24, 25	mpan 400	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On → ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On)
27	23, 26	syl6 29	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
28		onun2 4216	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On) → (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On)
29	8, 27, 28	syl6an 1323	. . . . . . . 8 ⊢ (𝜑 → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
30	29	adantr 261	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
31		rdgon.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn V)
32	31, 8	jca 290	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 Fn V ∧ 𝐴 ∈ On))
33		rdgivallem 5968	. . . . . . . . . 10 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ On ∧ 𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
34	33	3expa 1104	. . . . . . . . 9 ⊢ (((𝐹 Fn V ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
35	32, 34	sylan 267	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
36	35	eleq1d 2106	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (𝐴 ∪ ∪ 𝑥 ∈ 𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
37	30, 36	sylibrd 158	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (rec(𝐹, 𝐴)‘𝑧) ∈ On))
38	37	expcom 109	. . . . 5 ⊢ (𝑧 ∈ On → (𝜑 → (∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
39	38	a2d 23	. . . 4 ⊢ (𝑧 ∈ On → ((𝜑 → ∀𝑥 ∈ 𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On) → (𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
40	7, 39	syl5bi 141	. . 3 ⊢ (𝑧 ∈ On → (∀𝑥 ∈ 𝑧 (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On) → (𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
41	3, 6, 40	tfis3 4309	. 2 ⊢ (𝐵 ∈ On → (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ∈ On))
42	41	impcom 116	1 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306  Vcvv 2557   ∪ cun 2915  ∪ ciun 3657  Oncon0 4100   ↾ cres 4347   Fn wfn 4897  ‘cfv 4902  reccrdg 5956

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-irdg 5957

This theorem is referenced by:  oacl  6040  omcl  6041  oeicl  6042